Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).


f(x)=cos x+2√x / √x.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a function when the function approaches infinity or negative infinity as the input approaches a certain value from either the left or the right. This typically happens when the denominator of a rational function approaches zero while the numerator remains non-zero. Identifying vertical asymptotes is crucial for understanding the behavior of the function near those points.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, evaluating the left-hand limit (lim x→a^− f(x)) and the right-hand limit (lim x→a^+ f(x)) helps determine the behavior of the function as it nears the asymptote. This analysis is essential for understanding how the function behaves around points of discontinuity.

Rational Functions

A rational function is a ratio of two polynomials, which can exhibit vertical asymptotes when the denominator equals zero. In the given function f(x) = (cos x + 2√x) / √x, the denominator √x must be analyzed to find points where it becomes zero, leading to potential vertical asymptotes. Understanding the structure of rational functions is key to identifying their asymptotic behavior.